Originally posted by: Eeezee
Originally posted by: mariok2006
Originally posted by: RedArmy
Originally posted by: mariok2006
Originally posted by: Leros
Originally posted by: mariok2006
Oh man... Taylor Series. Good thing we didn't cover those in Calc II :roll:
You should have. Not too tricky to manipulate once you learn the basic ones that thesurge linked to.
Yeah, I know. The teacher said she couldn't cover it in time.
Talk about having a terrible teacher. They play a big role in a lot of different kinds of mathematics. It should be a requirement to teach at least something on the Taylor series, just to gain an understanding. It's not that easy if you're learning it for the first time.
Are they at all similar to Laplace transforms?
No, they're really not similar at all. They are kind of similar to Fourier series, however.
Wikipedia should have pretty much all you need to know on Taylor series. Check out the Definition section. Basically, you can express any infinitely differentiable function as an infinite sum of derivatives (and some constants and other terms are tacked on). The definition is all you need to know to derive pretty much any Taylor series.
Fairly frequently in upper division Physics courses you'll either be expressing a function as a taylor series and only taking the first few terms (a first-order or second-order approximation will only keep the first few terms, you can do this if the rest of the terms are becoming negligible) OR you'll have an infinite summation that looks like a Taylor series and you'll write down whatever function that is (this happens a lot in stat mech, where you have a near-infinite number of particles and thus infinite sums tend to pop up).
You have a fairly weird infinite sum there, however - the only term with a 1/2 is the x^2 and the rest of the terms have a 1/3?
My guess would be (it's just a guess, but it definitely satisfies the problem): ((1/(1-x))/3) - (1/3) - (x/3) - ((x^2) /3) + ((x^2) / 2)
The Taylor Series (Taylor Expansion) of 1/(1-x) is the sum of x^n, which is clearly what you have but with the first few terms missing. Easy enough, that just means we need to take the taylor series, subtract those first few terms, and then add the x^2 / 2 since it doesn't fit the rest of the series (which we forced to have a 1/3 under each term). That is definitely a correct answer, but I don't necessarily know if you've been learning some weird math functions that I've never heard of.
Don't worry, Taylor Series are really easy to learn. Once you have the definition and understand that it's true, it's just a hand crank. You can express nearly anything as a Taylor Series, but there are a few really easy functions (listed on the Wiki page) that are just handy to know. My Calc II teacher barely went over them, but it turns out that they're pretty important!
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